Variational inequalities are a formalism that includes games, minimization, saddle point, and equilibrium problems as special cases. Methods for variational inequalities are therefore universal approaches for many applied tasks, including machine learning problems. This work concentrates on the decentralized setting, which is increasingly important but not well understood. In particular, we consider decentralized stochastic (sum-type) variational inequalities over fixed and time-varying networks. We present lower complexity bounds for both communication and local iterations and construct optimal algorithms that match these lower bounds. Our algorithms are the best among the available literature not only in the decentralized stochastic case, but also in the decentralized deterministic and non-distributed stochastic cases. Experimental results confirm the effectiveness of the presented algorithms.

翻译：变分不等式是一个统一框架，其特例包括博弈、极小化、鞍点与均衡问题。因此，变分不等式方法适用于许多实际任务，包括机器学习问题。本文聚焦于日益重要但尚未充分理解的分布式场景。具体而言，我们研究了固定与时变网络上的分布式随机（求和型）变分不等式。我们给出了通信轮次与本地迭代次数的下界复杂度，并构建了匹配这些下界的最优算法。在现有文献中，我们的算法不仅在分布式随机情形下最优，在分布式确定性与非分布式随机情形下同样最优。实验结果验证了所提算法的有效性。